Prime Numbers

404 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
to suitable random numbers. But if one lifts the requirement of statistically
testable randomness as it is usually invoked, there is quite another way to
use random sequences. It is to these alternatives—falling under the rubric of
quasi-Monte Carlo (qMC)—to which we next turn.
8.3 Quasi-Monte Carlo (qMC) methods
Who would have guessed, back in the times of Gauss, Euler, Legendre, say,
that primes would attain some practical value in the financial-market analysis
of the latter twentieth century? We refer here not to cryptographic uses—
which certainly do emerge whenever money is involved—but quasi-Monte
Carlo science which, loosely speaking, is a specific form of Monte Carlo (i.e.,
statistically motivated) analysis. Monte Carlo calculations pervade the fields
of applied science.
The essential idea behind Monte Carlo calculation is to sample some large
continuous (or even discrete, if need be) space—in doing a multidimensional
integral, say—with random samples. Then one hopes that the “average” result
is close to the true result one would obtain with the uncountable samples
theoretically at hand. It is intriguing that number theory—in particular primenumber
study—can be brought to bear on the science of quasi-Monte Carlo
(qMC). The techniques of qMC differ from traditional Monte Carlo in that one
does not seek expressly random sequences of samples. Instead, one attempts to
provide quasirandom sequences that do not, in fact, obey the strict statistical
rules of randomness, but instead have certain uniformity features attendant
on the problem at hand.
Although it is perhaps overly simplistic, a clear way to envision the
difference between random and qMC is this: Random points when dropped can
be expected to exhibit “clumps” and “gaps,” whereas qMC points generally
avoid each other to minimize clumping and tend to occupy previous gaps. For
these reasons qMC points can be—depending on the spatial dimension and
precise posing of the problem—superior for certain tasks such as numerical
integration, min–max problems, and statistical estimation in general.
8.3.1 Discrepancy theory
Say that one wants to know the value of an integral over some D-dimensional
domain R, namely
I = ··· f(x) d
R
D x,
but there is no reasonable hope of a closed-form, analytic evaluation. One
might proceed in Monte Carlo fashion, by dropping a total of N “random”
vectors x =(x1,...,xD) into the integration domain, then literally adding up
the corresponding integrand values to get an average, and then multiplying
by the measure of R to get an approximation, say I ′ , for the exact integral
I. On the general variance principles of statistics, we can expect the error to